## Τρίτη, 28 Μαρτίου 2017

### TRIANGLE CENTERS

P001

Barycentrics: a (b-c) (a+b-c) (a-b+c) ((b-c) (-a+b+c) Sin[A/2]+(a-c) (a-b+c) Sin[B/2]+(-a+b) (a+b-c) Sin[C/2])::

lies on these lines: {65,2089},{177,10505},{1122,7 371},{6018,10508}

= X(7371)-Ceva conjugate of X(3669).

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25581

P002

Barycentrics: a (a^4 b+2 a^3 b^2-2 a b^4-b^5+a^4 c-6 a^3 b c+4 a^2 b^2 c-2 a b^3 c+3 b^4 c+2 a^3 c^2+4 a^2 b c^2-2 b^3 c^2-2 a b c^3-2 b^2 c^3-2 a c^4+3 b c^4-c^5-4 b c ((2 a^3+a^2 b-2 a b^2-b^3+a^2 c-4 a b c+b^2 c-2 a c^2+b c^2-c^3) Sin[A/2]-a ((a^2-b^2+6 a c+c^2) Sin[B/2]+(a^2+6 a b+b^2-c^2) Sin[C/2])))::

lies on these lines: {3,6585} and {1, P001}.

= X(7371)-Ceva conjugate of X(3669).

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25582

P003

Barycentrics: 11*SB*SC+7*S^2 : :

= 27*X(2)-11*X(3), 3*X(2)-11*X(5), 15*X(2)-11*X(140), 5*X(2)+11*X(381), 9*X(2)+11*X(546), 7*X(2)-11*X(547), 19*X(2)-11*X(549), 21*X(2)-11*X(3530), X(4701)+11*X(9955), X(5609)+3*X(11801)

= Shinagawa coefficients: (7, 11)

= midpoint of X(i) and X(j) for these {i,j}: {3,12102}, {4,3530}, {5,3850}, {140,3861}, {381,10109}, {546,3628}, {547,3860}, {3845,10124}, {5066,11737}

= reflection of X(i) in X(j) for these (i,j): (3856,3850), (11540,547), (11695,12046), (12108,3628)

= On lines: {2,3}, {517,4540}, {3303,10592}, {3304,10593}, {3614,3746}, {4701,5844}, {5418,10147}, {5420,10148}, {5563,7173}, {5609,11801}, {6488,8253}, {6489,8252}, {11695,12046}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (4,5,547), (4,547,3530), (4,632,12103), (4,3859,3860), (4,5070,8703), (4,5079,632), (5,3627,3090), (5,3845,1656), (140,12101,20), (381,3627,546), (546,3627,3861), (632,3627,8703), (1656,3845,548), (3090,3627,140), (3091,3146,3855), (3525,5076,550), (3628,12102,3), (3843,5056,549), (3850,3861,381), (3861,10109,140), (5055,5076,3525)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25590

P004

Barycentrics: 7*SB*SC+11*S^2 : :

= 27*X(2)-7*X(3), 3*X(2)+7*X(5), 12*X(2)-7*X(140), 13*X(2)+7*X(381), 2*X(2)-7*X(547), 6*X(2)-X(548), 17*X(2)-7*X(549), 15*X(2)-7*X(631), 9*X(373)+X(5876), 7*X(576)+3*X(3630)

= Shinagawa coefficients: ( 11, 7)

= midpoint of X(i) and X(j) for these {i,j}: {5,1656}, {140,3859}, {631,3858}, {632,3091}

= reflection of X(i) in X(j) for these (i,j): (546,3091), (632,3628), (3522,3530), (3843,3850), (5071,10109)

= On lines: {2,3}, {373,5876}, {576,3630}, {3303,10593}, {3304,10592}, {3614,5563}, {3625,10175}, {3633,5886}, {3635,5901}, {3746,7173}, {4668,5844}, {4691,9956}, {5305,7603}, {5690,7988}, {5943,12046}, {6560,10148}, {6561,10147}, {10095,10170}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2,3627,12108), (3,11541,550), (4,5,11737), (5,3627,5072), (5,3845,5068), (140,546,12103), (546,547,3628), (546,548,3627), (546,3091,3859), (546,3628,140), (546,12103,3853), (1656,3843,2), (3091,5076,3858), (3627,3850,546), (3627,5072,3850), (3627,12108,548), (3628,3856,10303), (3843,5072,3091), (3857,12102,546), (5070,12101,140), (10303,11541,3)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25590

P005

Barycentrics: a ((b-c) (a^4 b-a^3 b^2-a^2 b^3+a b^4+a^4 c+a^3 b c+a^2 b^2 c-a b^3 c-2 b^4 c-a^3 c^2+a^2 b c^2+2 a b^2 c^2+2 b^3 c^2-a^2 c^3-a b c^3+2 b^2 c^3+a c^4-2 b c^4)+b (b-c) c (2 a^3+a^2 b-2 a b^2-b^3+a^2 c+2 a b c+5 b^2 c-2 a c^2+5 b c^2-c^3) Sin[A/2]-a (c (a^3 b+2 a^2 b^2-a b^3-2 b^4+a^3 c-a^2 b c+5 a b^2 c+3 b^3 c-a^2 c^2-5 a b c^2+b^2 c^2-a c^3-3 b c^3+c^4) Sin[B/2]- b (a^3 b-a^2 b^2-a b^3+b^4+a^3 c-a^2 b c-5 a b^2 c-3 b^3 c+2 a^2 c^2+5 a b c^2+b^2 c^2-a c^3+3 b c^3-2 c^4) Sin[C/2]))::

on lines {{164,5708},{177,942},{5049, 8422},{5439,11691},{8083,8091} ,{9957,11191}},

See Peter Moses, Hyacinthos 25592

P006

Barycentrics: Sin[A/2] ((a-b-c) (a^2 b-b^3+a^2 c+2 a b c+b^2 c+b c^2-c^3) Sin[A/2]+2 a b c ((a-b+c) Sin[B/2]+(a+b-c) Sin[C/2]))]::

on lines {{1,7597},{57,3659},{65,2089}, {174,354},{177,942}}.

See Peter Moses, Hyacinthos 25592

P007

Barycentrics: = 4*a^4-6*(b^2+c^2)*a^2+5*(b^2- c^2)^2 : :

= On lines: {2,7765}, {4,5206}, {6,17}, {32,5056}, {115,140}, {187,3850}, {532,8260}, {533,8259}, {547,5007}, {550,3054}, {574,3533}, {629,6674}, {630,6673}, {1504,10195}, {1505,10194}, {3090,7753}, {3523,7756}, {3525,11648}, {3628,9698}, {3851,7747}, {5059,8588}, {5067,7772}, {5070,5309}, {5461,7824}, {6292,6722}

= midpoint of X(17) and X(18)

= reflection of X(i) in X(j) for these (i,j): (629,6674), (630,6673)

See Tran Quang Hung and César Lozada, Hyacinthos 25601

P008

Barycentrics: = 1/(S+5*SA) : :

= on Kiepert hyperbola and lines: {5,6434}, {372,3591}, {382,485}, {486,546}, {550,10195}, {1131,6561}, {1132,6436}, {1152,11737}, {1327,6470}, {1328,3070}, …

See Tran Quang Hung and César Lozada, Hyacinthos 25607

P009

Barycentrics: = 1/(S+5*SA) : :

1/(-S+5*SA) : :

= on Kiepert hyperbola and lines: {5,6433}, {371,3590}, {382,486}, {485,546}, {550,10194}, {1131,6435}, {1132,6560}, {1151,11737}, {1327,3071}, {1328,6471},…

See Tran Quang Hung and César Lozada, Hyacinthos 25607

P010

Barycentrics: = 1/(S+5*sqrt(3)*SA) : :

= on Kiepert hyperbola and lines: {17,382}, {18,546}, {383,11669}, {550,10188},…

See Tran Quang Hung and César Lozada, Hyacinthos 25607

P011

Barycentrics: 1/((3*(1+sqrt(2)))*SA+S) : :

= on Kiepert hyperbola and lines: {30,3373}, {381,3388},…

See Tran Quang Hung and César Lozada, Hyacinthos 25607

P012

Barycentrics: 1/((3*(1+sqrt(2)))*SA-S) : :

= on Kiepert hyperbola and lines: {30,3388}, {381,3373},…

See Tran Quang Hung and César Lozada, Hyacinthos 25607

P013

Barycentrics: a^2 (a^4-b^4+b^2 c^2-c^4) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)::

on lines {{2,2781},{23,6593},{25,110},{ 51,542},{52,10294},{74,9818},{ 113,403},{125,5133},...}}.

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609

P014

Barycentrics: a^2 (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+7 a^4 b^2 c^2-4 a^2 b^4 c^2-b^6 c^2-4 a^2 b^2 c^4+4 b^4 c^4+2 a^2 c^6-b^2 c^6-c^8)::

on lines {{2,974},{3,74},{22,9934},{69, 146},{113,403},{125,5907},...} }.

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609

P015

Barycentrics: a (a+b) (a+c) (a^4-b^4+a^2 b c-a b^2 c-a b c^2+2 b^2 c^2-c^4) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)::

on lines {{21,2778},{28,110},{113,403}, ...}}.

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609

P016

Barycentrics: (a^4 b^2-b^6+a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+b^2 c^4-c^6) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)::

on lines {{2,98},{5,12099},{113,403},.. .}}..

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609

P017

Barycentrics: (2 a^2-b^2-c^2) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)::

on lines {{4,541},{25,542},{51,125},{10 7,11005},{110,6353},{112,6792} ,{113,403},...}}.

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609

P018

Barycentrics: (a^2-b c) (a^2+b c) (2 a^4-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4)::

on lines {{6,98},{32,2782},{39,12042},{ 99,3053},{114,230},{115,546},. ..}}.

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609

P019

Barycentrics: (a^4+a^2 b^2-b^4+a^2 c^2-b^2 c^2-c^4) (2 a^4-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4)::

on lines {{6,147},{30,1569},{98,3815},{ 99,7762},{114,230},{115,3850}, ...}}.

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609

P020

Barycentrics: (2 a^2-a b-b^2-a c+2 b c-c^2) (a^3 b-a^2 b^2-a b^3+b^4+a^3 c+a b^2 c-a^2 c^2+a b c^2-2 b^2 c^2-a c^3+c^4)::

on lines {{11,118},{12,5884},{57,5660}, {63,3035},{80,11529},{100,3474 },{119,912},...}}.

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609

P021

Barycentrics: (2 a-b-c) (a+b-c) (a-b+c) (a^3 b-a^2 b^2-a b^3+b^4+a^3 c+a b^2 c-a^2 c^2+a b c^2-2 b^2 c^2-a c^3+c^4)::

on lines {{1,6713},{10,5083},{11,65},{1 2,5883},{46,5840},{56,952},{57 ,80},{78,3035},{100,1788},{104 ,1470},{109,6788},{119,912},.. .}}.

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609

P022

Barycentrics: a^2 (a-b) (a+b) (a-c) (a+c) (a^4 b^4-2 a^2 b^6+b^8-2 b^6 c^2+a^4 c^4+4 b^4 c^4-2 a^2 c^6-2 b^2 c^6+c^8)::

on lines {{4,69},{99,512},{112,249},... }}.

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 25609

P023

Trilinears: (a^4-3*(b^2+c^2)*a^2-7*b^2*c^ 2+2*c^4+2*b^4)*a : : = On lines: {2,576}, {6,11451}, {22,10541}, {25,5012}, {51,5092}, {110,5943}, {140,1173}, {182,5645}, {184,10545}, {186,5462}, {323,6688}, {373,1994}, {589,8956}, {597,11416}, {694,3108}, {1350,3060}, {1597,10574}, {2979,5644}, {3567,7514}, {3580,11548}, {5020,11422}, {5133,9140}, {5899,10095}, {9781,12083}, {9815,20009}, {10546,11402}

= {X(5422), X(5640)}-Harmonic conjugate of X(5012)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 25612